ALGEBRAIC AND GEOMETRIC COMPLEXITY THEORY

cs598maf (SPRING 2023)

The use of algebraic techniques (addition, multiplication, derivatives, and beyond) is pervasive in the design of efficient computation. This course will explore how these techniques apply to problems of an inherently algebraic nature, such as computing the determinant of a matrix, as well as for tasks that seemingly lack algebraic structure, such as combinatorial optimization.

The course will develop the theory of using algebraic circuits to compute multivariate polynomials. This includes non-trivial algorithmic paradigms for efficient computation in this model (upper bounds), methods for proving that certain polynomials cannot be efficiently computed (lower bounds), as well as algorithms for efficiently deciding whether a given algebraic circuit computes the zero polynomial (polynomial identity testing).

The course will also explore the underlying algebraic geometry of algebraic computation, especially in order to address foundational questions. This exploration will be with an eye toward the Geometric Complexity Theory program of Mulmuley and Sohoni, which aims to resolve the algebraic version of the P vs NP question.

**Lecture:** TR2-3:15, Digital Computer Laboratory 1310.

**Staff:**

role | name | (netid) |
office hour |
---|---|---|---|

instructor: |
Prof. Michael A. Forbes | (miforbes) |
T3:30, Siebel 3220 |

The **calendar** lists lecture topics, associated lecture materials, and auxiliary reading. It also lists the problem sets with release- and due-dates.

The forum for the class is Piazza (code).

The submission server for the class is Gradescope (code). When submitting assignments, student must use their `@illinois.edu` netid so the course staff can appropriately map students to submissions.

A students grade in the course compromises the 6 psets of 4 problems each.

The lectures for the course will be in-person.

Materials from lecture (slides, boardwork, etc) will be posted, generally within 24 hours of the lecture, on the calendar. Recordings of the lecture will similarly be posted, generally within 24 hours of the lecture, on the calendar.

The main materials for the class are the above-mentioned lecture materials. Auxiliary pointers to written material will be listed for each lecture, and are approved materials for collaboration purposes. Due to the advanced nature of the course, the pointed-to materials will often significantly deviate in presentation from the lecture.

The following (approved) materials will generally relevant to the course, and will be referenced by their indicated abbreviation

*Arithmetic Circuits: A Survey of Recent Results and Open Questions*, by Amir Shpilka and Amir Yehudayoff [SY]*Algebraic Complexity Theory*, by Bürgisser, Clausen, and Shokrollahi [BCS]*Completeness and Reduction in Algebraic Complexity Theory*, by Bürgisser*Workshop on Algebraic Complexity (2016)*[WACT16]*Polynomial identity testing of read-once oblivious algebraic branching programs*, by Michael A. Forbes*A survey of lower bounds in arithmetic circuit complexity*, by Saptharishi*Partial Derivatives in Arithmetic Complexity and Beyond*, by Chen, Kayal, and Wigderson [CKW]*Ideals, Varieties, and Algorithms*, by Cox, Little, and O'Shea [CLO]

Familiarity with undergraduate algebra, complexity theory, and randomized algorithms, as well as mathematical maturity.

The course will use Piazza as a forum.

The forum will be the sole source of announcements from the course staff. Students should sign-up promptly to avoid missing such announcements.

The forum is also intended for students to connect amongst themselves on all course-related issues, **except** when such discussion reveals significant information about solutions to pending assignments. Posts can be made anonymously to other students (but not to the course staff). The students in the course can sometimes offer the best help; please make this resource the best it can be.

The forum is finally meant for private communication between the students and the course staff, especially for logistical matters such as scheduling conflicts. Emailing course staff directly is *heavily discouraged*. While the forum can be used for private help from the course staff on course content, a quicker and more thorough response is often achieved through a public discussion.

Coursework must be submitted individually. However, verbal discussion (oral or electronic) with other students in the course is highly encouraged, subject to the constraint that submitted psets must list all such collaborators. However, the composition of the pset submissions must be solely the work of the listed author. In particular, no sharing of written solutions is allowed.

Students are allowed to use any listed course materials, or discussions with students or staff affiliated with the course. Students are forbidden from using any other online, textbook, or human resource.

The first violation of academic integrity is a zero for the entire assignment. The second violation is a reduction of 1 letter grade.

A students grade in the course compromises the 6 psets of 4 problems each. All problems from problem sets are worth the same number of points.

Regrade requests will be open for two weeks after an assignment is returned. Regrade requests will result in the *entire* assignment in being regraded; as a result the overall score may decrease.

Sample solutions will be distributed to students when the problem sets are returned (please keep the internet free of easily-found solutions!). These sample solutions will be selected from student submissions (with names omitted). Please inform the course staff if you wish to * opt-out* of ever being selected.

Students are highly encouraged to turn-in assignments on-time to avoid falling behind on the material, and to incentivize this any late problem sets will automatically lose 10% in value. However, to be flexible, for each pset students can automatically take (without asking) a 3-day extension (72 hours). The extension (and resulting penalty) applies to the *entire* problem set, regardless of whether a partial submission was made on time. Problem sets will not be accepted past this 3-day extension window.

Figures are **highly** encouraged. Many algorithms act on combinatorial objects such as graphs, and discussing such objects without a figure is a pain to write, and to read.

To use your time wisely, students are encouraged to focus on *content* of their coursework over *format*. However, content and format are not fully independent. As such, students are suggested to consider the following advice on submission formats.

**typesetting submissions:**Typesetting is generally encouraged. Electronic documents are easier to revise, and as such easier to use to ensure content quality as first drafts are rarely polished. Typesetting ensures legibility, which is important as the course staff cannot give credit for work they cannot read. Finally, learning to typeset is a skill useful in many parts of academia and industry.**However**, including figures in typeset documents is often difficult, to the point that often students prefer to omit figures altogether. This can lead to a significant decrease in coursework clarity and concision, which can impact the score received. For beginners to typesetting there is the additional burden of having to learn to typeset while learning the course material. The result is often an ungainly document, to the point that the legibility gained by typesetting is offset by the poor usage of typesetting's abilities.**handwriten submissions:**Handwritten submissions are accepted, as long as they are at least as legible as the handwriting of the instructor, and also that the submissions use**figures**.